The Finite Basis Problem for Multiplicatively Idempotent Semirings

Introduction

We ask whether every multiplicatively idempotent semiring has a finite identity basis in the signature $\{+, \cdot\}$. A semiring is an algebra $(S,+,\cdot)$ in which addition is associative and commutative, multiplication is associative, and multiplication distributes over addition from both sides. We call a semiring multiplicatively idempotent if it satisfies $x^2 \approx x$, and write MI for the variety of all such semirings. Neither an additive identity nor a multiplicative identity is part of the signature, and we do not assume commutative multiplication. For a positive integer $n$, the notation $nx$ denotes a sum of $n$ copies of $x$.

An algebra is finitely based if its identities follow from a finite set of identities. A variety is hereditarily finitely based if each of its subvarieties is finitely based. Additively idempotent semirings can be nonfinitely based even when their multiplicative reducts are finitely based; see Jackson, Ren and Zhao [6]. Ren, Yue and Zhou [11] investigate inherent nonfinite basability in the additively idempotent setting.

2020 Mathematics Subject Classification. Primary 16Y60; Secondary 08B05, 20M07, 03C05.

Key words and phrases. Multiplicatively idempotent semiring, finite identity basis, hereditary finite basability, band, semiring variety.

This work was supported by the Science and Technology Research Program of Chongqing Municipal Education Commission (KJZD-K2024011102) and the Chongqing Natural Science Foundation Innovation and Development Joint Fund (Municipal Education Commission) (CSTB2025NSCQ-LZX0067).

For multiplicatively idempotent semirings, Vechtomov and Petrov [12] proved that MI is locally finite and that, when multiplication is commutative, its members decompose subdirectly into semirings satisfying $3x \approx x$ and $3x \approx2x$. We establish this decomposition for noncommutative multiplication as well.

In the commutative setting, Chajda and Länger [2, 3] studied multiplicatively idempotent semirings with designated constants 0 and 1. They obtained a finite basis and polynomial normal forms for the variety generated by the three-element semiring $T_3$ [3]. Its $\{+,\cdot\}$-reduct occurs below as an obstruction to the identity $2(x+y) \approx x+y$. We work with this reduct in the constant-free signature.

Wang and Shao [13 Corollary 3.4] proved hereditary finite basability for the semiring variety determined, for each integer $n \ge2$, by

At $n=2$, this theorem applies to the subvariety of MI satisfying $x+2xyx \approx x$. The reduct of $T_3$ used below fails this absorption identity. This failure shows that the theorem does not cover all of MI.

Every variety of bands is finitely based [4]. The variety of semirings in which both operations are idempotent has 78 subvarieties, all finitely based [5, 8, 10]. These results do not by themselves control the undoubled summands of a semiring polynomial. An arbitrary band can occur as a multiplicative subsemigroup after adjoining an absorbing element and making addition constant. The multiplicative reducts of these semirings need not satisfy a regular-band identity.

We prove the following theorem.

The defining axioms give a finite basis for MI itself. Theorem 1.1 asserts finite basability of every subvariety, a conclusion that does not follow from local finiteness.

For the subvariety satisfying $3x \approx2x$, we express each polynomial identity using a common doubled part and isolated word occurrences. We apply the theory of doubly idempotent semirings to the doubled part; the isolated word occurrences retain the original band structure. Four fixed identities align their first- and last-occurrence words. Once these words agree, recursive extraction gives an identity of the form

where $\omega$ is a fresh variable. The associated word identities form a fully invariant band congruence. The finite basis theorem for bands applies to the resulting band variety.

Preliminaries

We follow the terminology of universal algebra in [1]. For an algebra $S$, let $\operatorname{Var}(S)$ be the variety generated by $S$. For a variety $W$, let $\operatorname{Id}(W)$ be its identity theory. We take bases for semiring subvarieties relative to the following finite set $\mathrm{SR}$:

We have $\mathrm{MI}=\operatorname{Mod}(\mathrm{SR},x^2\approx x)$. We write identities with $\approx$ and use equality for literal words and for elements under a fixed evaluation.

Fix a countably infinite alphabet. A word is a nonempty multiplicative word, and $c(w)$ is its content. Let $\ell(w)$ and $r(w)$ denote the first-occurrence word and last-occurrence word of $w: retain, respectively, the first and the last occurrence of each variable in its original left-to-right order. A polynomial is a finite nonempty sum of word occurrences. We count occurrences with multiplicity, so $u+u$ has two occurrences. An occurrence is supported in a set of variables when its content is a subset of that set. Every semiring term expands into such a polynomial. Missing contexts and omitted sums are notation only; we never substitute an empty word or an empty sum for a variable.

Write $\mathrm{B}$ for the variety of bands, that is, idempotent semigroups, and $\mathrm{B}X$ for the free band on a finite nonempty alphabet $X$. Every subvariety of $\mathrm{B}$ has a finite basis [4]. The structure theorem for bands [7] gives the following deletion rule.

For a word $w$ involving at least two variables, let $s(w)$ be its longest prefix whose content is a proper subset of $c(w)$, and let $e(w)$ be its longest suffix with proper content. These are actual factors of $w$, possibly with repeated variables. If $a$ is the last letter of $\ell(w)$ and $b$ is the first letter of $r(w)$, then

Vechtomov and Petrov [12 p. 54 of the Russian original] observed that $\mathrm{MI}$ is locally finite. The proof below bounds the number of normal forms used later.

This bound counts polynomial representatives; distinct coefficient vectors may define the same term function.

An algebra in IB is called doubly idempotent. We use three results:

• (i) every subvariety of IB is finitely based;

• (ii) every multiplicative reduct of a member of IB is a regular band;

• (iii) in the variety RB of regular bands,

Here $\mathrm{RB} = \operatorname{Mod}(x^2 \approx x,\, xyxzx \approx xyzx)$. Statement (i) follows from [5, 8]; see [10 Result 4 and Theorem 7]. Pastijn and Zhao [9, Theorem 2.3] proved (ii), and Polák [10, Result 1] recalls the word criterion in (iii). We include a verification of (ii).

Proof. In a doubly idempotent semiring, put $a \le b$ when $a+b=b$. Multiplication is monotone, and $(u+v)^2=u+v$ implies $uv\le u+v$. Multiplying this inequality on the left and right gives

Let $p=xyxzx$, $q=xyzx$, and $t_0=xyzyxzx$. The following applications of (2.5) use only band identities to simplify their entries:

The three rows give $q\le p$, $t_0\le q$, and $p\le t_0$, proving $p=q$. Apply this result to the doubled image in Lemma 2.3.

Structural reductions

Put

Vechtomov and Petrov [12 Proposition 3.1] proved this decomposition for commutative multiplication. We extend it to arbitrary members of MI and to each subvariety.

Proof. Define $a\preceq b$ if $a=b$ or $b=a+u$ for some $u\in S$, and let $H_+$ be the symmetric part of this preorder. This is the Green relation of the commutative additive semigroup. Every additive translation and every left or right multiplicative translation preserves $\preceq$, by distributivity. This translation invariance makes $H_+$ a semiring congruence. As $3a=2a+a$ and $2a=3a+a$, its quotient satisfies $3x\approx2x$.

We claim that $H_+\cap\ker r$ is the equality relation. Suppose $aH_+b$ and $a\ne b$. Write $b=a+u$ and $a=b+v$. Iterating $a=a+u+v$ gives $a=a+2u+2v$. By $4u=2u$,

With $c=3u$, we obtain $3c=c$, $b=a+c$, and $a=b+c$. If $3a=3b$, then $3a+c=3a$. Adding $a$ gives $e+c=e$, where $e=2a$. Put $p=ac$. Right multiplication by $c$ yields $2p+c=2p$, and $3p=p$ gives $p+c=p$. Expanding the square gives

which gives $b+c=b$. Since $a=b+c$, this contradicts $a\ne b$. This proves the claim and the embedding (3.1). Its two coordinates are surjective homomorphisms. The factors are homomorphic images of $S$; applying the decomposition to every member of $W$ proves (7).

The branch $3x \approx x$

Let $\mathrm{BR}$ be the variety of Boolean rings in the signature $\{+, \cdot\}$. The identities $\mathrm{SR}$, $x^2 \approx x$, $2x \approx2y$, and $3x \approx x$ define this variety. The common value of $2x$ is the zero element of the ring, although it is not a constant symbol.

For $S \in\mathrm{R}$ and $e \in d(S)$, the set

is a subsemiring. Its addition is a group of exponent at most two, with identity element $e$, since $a + e = 3a = a$ and $a + a = e$. Distributivity and multiplicative idempotence make it a Boolean ring. Its multiplication is commutative: expanding $(a + b)^2 = a + b$ and cancelling in the additive group gives $ab + ba = e$ and $ab = ba$.

For a polynomial $P$, define $\pi(P)$ to be the set of nonempty contents occurring an odd number of times among its word occurrences. In Boolean rings, a word depends only on its content, and distinct square-free polynomials over the two-element Boolean ring define distinct functions. This uniqueness gives the criterion

The empty parity set records a zero value; it does not denote an empty sum term.

The branch in which every sum is idempotent

Put

The latter is the variety of constant-addition band semirings. The common sum is a multiplicative absorbing element, by distributivity. Every term containing addition has that same value.

Proof. For $S \in\mathbf{J}$, let $D=d(S)$. Every sum belongs to $D$, and $D$ is a multiplicative ideal since $2(es)=(2e)s=es$ and $2(se)=s(2e)=se$ for $e \in D$. The relation $\rho_D$ which collapses $D$ to one class and leaves other elements separate is a semiring congruence. Its quotient has constant addition. The map

is a subdirect embedding: the first coordinate separates any two elements in the collapsed class, and the second separates any pair not both in $D$. Applying this embedding to every $W \subseteq\mathbf{J}$ gives

Apply Lemma 3.2, Lemma 3.5, and the hereditary finite basability of $\mathrm{IB}$. $\square$

Absorption and polynomial normal forms

For the subvarieties of $\mathbf{A}$ outside $\mathbf{J}$, we need normal forms for polynomial identities. The additive identity $3x\approx2x$ lets us remove absorbed summands one at a time without cancellation.

Proof. If $s+a+b=s$, iteration gives $s+2a+2b=s$, and

This proves (i). Expanding $(x+y)^2=x+y$ gives

and (i) gives (ii). For (iii), associativity gives additive closure. If $a,b\in I_s$, then

and (i) gives $ab\in I_s$. Multiplying $s+a=s$ on either side by $s$ proves the remaining assertions. $\square$

Substituting $y=x$ in assertion (ii) gives $3x\approx2x$. This substitution proves that the absorption identity characterizes $\mathbf{A}$ inside $\mathbf{MI}$.

A three-element obstruction

Let $\mathsf{T}$ be the three-element semiring with the following tables:

+	e	s	f	·	e	s	f
e	e	e	e	e	e	e	f
s	e	e	s	s	e	s	f
f	e	s	f	f	f	f	f

The symbols $e$, $s$, $f$ are element names. The tables give $\mathsf{T} \in\mathcal{A} \setminus\mathcal{J}$. Chajda and Länger introduced the semiring $\mathsf{T}_3$ in [2 Definition 4.1] and studied it in [3]; $\mathsf{T}$ is its $\{+,\cdot\}$-reduct. In the notation $\mathsf{T}_3 = \{0,a,1\}$ of [3], the following map is an isomorphism:

The symbols in these images name elements of the cited algebra; they do not add constants to the present signature.

Proof. Choose $a,b \in S$ with $s = a+b \ne2s$, and put

By Lemma 4.1, $s$ absorbs $ab$ and $ba$ and their products $aba$ and $bab$. These absorptions give $s+f=s$. Expansion using band identities gives

We have $es = se = e$, $ef = fe = f$, $s+e=e$, $e+f=e$, and $2f=f$. Since $s \ne2s$, it is distinct from $e$ and $f$. If $e=f$, the equations $s+f=s$ and $s+e=e$ imply $s=e$, a contradiction. The resulting tables on $\{e,s,f\}$ are (4.1). The final assertion follows from Proposition 3.6. □

In $\mathsf{T}$, taking $x=y=s$ gives $x+2xyx=3s=e \ne s$. This evaluation proves that the absorption identity required in [13] at $n=2$ does not follow from multiplicative idempotence.

For a polynomial $P$ on a finite alphabet $X$ and a subset $D \subseteq X$, define

An occurrence $u$ in $P$ is isolated if $\nu_P(c(u))=1$. We call its content an isolated content of $P$ and denote the set of these contents by $\mathcal{I}(P)$. To distinguish reduced polynomials, Chajda and Länger [3 Theorem 4.1] used assignments to 0 and 1. The evaluations below adapt those assignments to the constant-free signature. The truncated count records how many occurrences have content in a given set, up to a maximum of two, for a possibly noncommutative polynomial.

Lifting doubled absorption

The two occurrences in Lemma 4.4 may be equal as words, but both copies must already be present. We cannot duplicate a single isolated occurrence.

Let $V \subseteq\mathbb{B}$ and let $P$ be a polynomial on a fixed finite alphabet $X$. Choose one word representative for each element of $B_X$ and put

Content and the absorption condition are independent of the representatives.

where $u_D$ is the unique isolated occurrence with content $D$. If there are no isolated occurrences, we write only $C_P$.

If $P \approx Q$ holds in a variety $\mathcal{W} \subseteq\mathcal{A}$ containing $\mathbf{T}$, and $\mathcal{V} = \mathcal{W} \cap\mathsf{IB}$, then, using the same ambient alphabet, the two sets in (4.4) coincide. We may use the same doubled polynomial $C$ for $P$ and $Q$.

Proof. For $q \in D_{\mathcal{V}}(P)$, the doubled image condition implies $\mathcal{Z} \models2P + 2q \approx2P$. Lemma 4.4 gives $P + q \approx P$ and $P + 2q \approx P$. Each nonisolated occurrence $w$ of $P$ belongs to $D_{\mathcal{V}}(P)$: its count is 2, and $P$ absorbs it in every additively idempotent semiring. We add all the doubled summands and remove each nonisolated original occurrence using $w + 2w \approx2w$. This proves (4.5).

To prove nonemptiness, take the product of all the word occurrences of $P$. In a doubly idempotent semiring, $P$ absorbs every occurrence and their product by Lemma 4.1. The product contains the content of every occurrence; its count is 2. It lies in (4.4).

For the final assertion, Lemma 4.3 gives $\nu_P = \nu_Q$ and common isolated contents. Since $\mathcal{V} \models P \approx Q$, the absorption conditions in (4.4) agree. □

Adding the doubled words in (4.5) preserves $\nu_P$ and all isolated contents: if $q \in D_{\mathcal{V}}(P)$ and $c(q) \subseteq E$, then $\nu_P(E) = 2$ already.

Absorption of words with full content

We use Polák’s description of $\mathsf{IB}$ [10] to test absorption of full-content words. For a polynomial $R$ on $X$, a letter $a \in X$, and $E \subseteq X$, put

if some occurrence $w$ of $R$ contains $a$ and all letters preceding the first $a$ in $w$ lie in $E$. This condition is monotone in $E$. For $i \in\{0,1,2\}$, we call a repetition-free word $a_1 \cdots a_k$ left admissible at level $i$ for $R$ under the following conditions:

• (0) there is no restriction;

• (1) if $k > 0$, the letter $a_1$ is the first letter of some occurrence of $R$;

• (2) $\rho_R(a_j,\{a_1,\ldots,a_{j-1}\})=1$ for every $j$.

The empty sequence is admissible at each level. Reversing all the words defines right admissibility. The empty sequence belongs to this combinatorial definition and does not introduce a semiring term.

Proof. By [10 Result 4], every subvariety of $\mathsf{IB}$ is generated by a subset of the eleven algebras

in that paper. The algebras $L$ and $R$ have two-element chain addition and left-zero or right-zero multiplication; $D$ is the two-element distributive lattice; and $M$ has both operations equal to chain join. The algebra $B$ is a two-element left-zero band with an identity element adjoined, ordered as $a<1<b$, and $C$ is its right-zero dual. Addition is join. The superscript $0$ adjoins an element which is an additive identity and a multiplicative zero. These descriptions specify elements, without adding constants to the signature.

Polák’s absorption tests [10 Result 3] say that $L$ tests the first letter and $R$ the last letter. The algebra $B$ tests, for every set $Y$ of deleted letters, whether the first letter of $g$ after deleting $Y$ is the first letter of some occurrence of $R$ after the same deletion; $C$ gives the reverse test. For $D$, $M$, $M^{0}$, the condition $\mathrm{c}(g)=\mathrm{c}(R)=X$ satisfies the absorption tests. The test for an extension by $0$ restricts $R$ to occurrences supported in $\mathrm{c}(g)$, which here retains all occurrences. The algebras $L$, $L^{0}$ impose left level 1; $B$, $B^{0}$ impose left level 2; their duals impose the corresponding right levels.

To identify the test for $B$ with level 2, write $\ell(g)=a_{1}\cdots a_{n}$. Necessity follows by deleting $\{a_{1},\ldots,a_{j-1}\}$ for each $j$. For sufficiency, let $Y\subsetneq X$ be any proper subset and let $a_{j}$ be the first letter of $g$ outside $Y$. Its predecessors in $\ell(g)$ lie in $Y$. Monotonicity in (14) supplies an occurrence of $R$ whose first surviving letter is $a_{j}$. Deleting all of $X$ gives equal empty words. Take the strongest left and right requirements among the chosen generators. These determine $\lambda,\mu$ and prove the lemma, including the trivial variety, for which both levels can be 0.

Proof. At level 2, a letter of $q$ absent from $p$ has at least the same predecessor set in $\ell(pq)$ as it had in $q$. Monotonicity of (14) preserves its admissibility. Prefix closure and the assertion for an occurrence follow from the definition. At level 0 there is no restriction; at level 1 the only condition concerns the initial letter.

Protected identities and recursive extraction

The normal forms leave isolated word occurrences that the doubled part does not determine. To compare them, we introduce protected identities. The next results apply to arbitrary subvarieties of $\mathrm{ML}$ and do not require the additive identity $3x\approx2x$.

Definition 5.1. For $W\subseteq\mathrm{ML}$, define a relation $\mathcal{G}_{W}$ on nonempty words by

where $\omega$ is a variable not occurring in $u,v$. We call the displayed semiring identity the protected identity associated with $u\approx v$.

The relation $\mathcal{G}_W$ is a theory of word identities; $+$ occurs only in its definition. We call it the protected band theory of W.

in the required order. These matching orders give $2t\omega_t \approx2h$, which turns the last displayed identity into the protected identity for $u,t$. □

Proof. We induct on $k = \lvert D\rvert$. If $k = 1$, both words are positive powers of the same variable and agree in every band. Suppose $k \ge2$ and assume the result for smaller contents. Write

Set $L = s(u)$, $R = e(u)$, $S = s(v)$, and $F = e(v)$. Lemma 5.3 gives

We show that $Lv \equiv_{G_W} v$ and $vR \equiv_{G_W} v$.

Choose $d \in D \setminus\{a_k\}$ and let $\alpha$ replace $a_k$ by $d$, fixing all other variables, including those outside $D$. The transformed isolated words have common content $E = D \setminus\{a_k\}$. Every other occurrence still contains a variable outside $D$ and outside $E$. To obtain the first-occurrence word after this identification, apply $\alpha$ to the former first-occurrence word and retain the first occurrence of each resulting letter. The analogous procedure, retaining last occurrences, gives the last-occurrence word. Both procedures preserve the equalities in (5.2). Since $\lvert E\rvert= k - 1$, the induction hypothesis applies and gives

The substitution fixes $L$ and $S$, both of content $E$. The word $u\alpha$ begins with $L$, the word $v\alpha$ begins with $S$, and both use only letters of $E$. Lemma 2.1 gives

Multiplying (5.5) on the right by $S$ within the congruence yields $LS \equiv_{G_W} S$. Write $v = Sv_1$, where $v_1$ is its actual nonempty remaining suffix, and append $v_1$. We obtain

For the dual argument, identify $b_1$ with any other letter of $D$, fixing all outside variables; call this substitution $\beta$. The induction hypothesis gives $u\beta\equiv_{G_W} v\beta$. The substitution fixes $R$, $F$, which have content $D \setminus\{b_1\}$. Left multiplication by $F$ and Lemma 2.1 give

We obtain $FR \equiv_{G_W} F$. Prepend the actual nonempty prefix $v_0$ in $v = v_0F$ to get

Combining (19)–(22) in the congruence gives

The induction preserves isolation by identifying variables only within the isolated content. It derives a protected identity without cancelling the doubled part of the original polynomial.

Four identities for occurrence-order alignment

Let $\Pi$ be the set of the six permutations of $y,z,w$. Define

We call the words in these doubled sums guard words and the sums guard polynomials. They occur in the following two guarded identities:

Reversing each multiplicative word gives the guarded identities $(H_r)$ and $(J_r)$. The four guarded identities involve at most four variables. Every guard polynomial in (23) has exactly six displayed doubled summands.

Proof. For (i), let $a$ be the first letter of $v$. Replace $a$ by $z$, every other letter of $D$ by $y$, and every outside letter by $w$. Multiply the identity on the right by $yz$. Both isolated occurrences have content $\{y,z\}$; one begins with $y$, the other with $z$. A word over two generators in a band is determined by its content and its first and last letters. Deleting consecutive repeated letters leaves alternating words, and those of length at least four reduce using $(yz)^2\approx yz$ and $(zy)^2\approx zy$. The two isolated occurrences become $yz$ and $zyz$.

Each other occurrence contains an outside letter and, after multiplication by $yz$, has content $\{y,z,w\}$. It is nonisolated: the isolated occurrence and this occurrence both have support in that content. Equation (3) shows that the doubled transformed polynomial absorbs the double of this occurrence. Lemma 4.4 permits replacing the occurrence by its double. Its last-occurrence word is $wyz$. By (4), its double is one of the six terms $2\pi yz$ in $C_h$. Adding $C_h$ to the transformed identity absorbs all these doubled terms on both sides and yields (6.2). For (ii), let $p_0$ be the longest common prefix of $\ell(u)$ and $\ell(v)$, and let $a$ be the next letter in $\ell(v)$. Put $D_0=c(p_0)$. Replace letters of $D_0$ by $x$, the letter $a$ by $Z=xzx$, all other letters of $D$ by $Y=xyx$, and all outside letters by $W=xwx$. Multiply on the left by $x$ and on the right by $ZY$.

The values $x,Y,Z,W$ lie in the local band $xSx$, with identity element $x$. Here $x$ is a nonempty term, not a constant symbol. After we omit this local identity from the calculation, the two isolated occurrences are words over $Y,Z$; one begins with $Y$, the other with $Z$, and both contain both letters. Appending $ZY$ gives the values $YZY$ and $ZY$. In the original variables, these are the undoubled words in (25).

Every remaining occurrence has full content $\{x,y,z,w\}$, first letter $x$, and last-occurrence word $wzyx$. We double it by applying Lemma 4.4 to the isolated occurrence and this occurrence. Each doubled word is one of $2x\pi zyx$. Adding $C_j$ yields (25). Reversal gives the dual assertions. □

Proof. For (i), Lemma 2.1 and band idempotence give

For (ii), the actual prefix relation gives $pu \approx u$. Substitution and right multiplication by $u$ give the two undoubled words

For the first reduction, simplify $pu \approx u$ repeatedly and use $upau \approx u$ since $c(pa) \subseteq c(u)$. The second reduction follows by successive uses of $pu \approx u$.

To check absorption of the guard words, apply Lemma 4.6. In (i), their first-occurrence words come from the six orders of $u,a,g$ after deleting repetitions. In (ii), they come from

The first-occurrence words of $u$ and are $\ell(u)$ and $a$. Lemma 4.7 gives admissibility of $\ell(pau)$ and permits completing each prefix with the admissible full word $\ell(g)$. Placing $g$ earlier in the concatenation also preserves admissibility. This verifies all six words in (26) and the six words in (i).

All six guard words in either case have the same last-occurrence word: delete the letters of $D$ from $r(g)$ and append $r(u)$. A full copy of $u$ follows every occurrence of $g$, while all other inserted factors have content in $D$. After reversal, retaining first occurrences in the concatenation $\operatorname{rev}(u)\operatorname{rev}(g)$ gives the corresponding first-occurrence word. The right-hand version of Lemma 4.7 proves its admissibility. The factor $g$ supplies full content. Lemma 4.6 proves absorption of every guard word. □

Here $p$ must be an actual prefix of $u$. Its repetition-free first-occurrence word need not satisfy $pu \approx u$ in an arbitrary band.

Proof. Deleting repetitions from $\ell(u)\ell(g)$ gives the first-occurrence word of $ugu$. The same operation on $\ell(\operatorname{rev}(u))\ell(\operatorname{rev}(g))$ gives its reversed last-occurrence word. Apply Lemmas 4.7 and 4.6. Replacing $u$ by $v$ gives the proof for $vgv$. □

Finite bases and the main theorem

For a word identity $\sigma=(s\approx t)$ write

where $\omega$ is fresh. For a polynomial identity $\tau=(P\approx Q)$ write $2\tau$ for $2P\approx2Q$. These are abbreviations for identities in the original two-operation signature.

Proof. The finite bases exist by the finite basis theorems for bands [4] and $\mathbf{IB}$ [5, 8], and by Lemma 5.2. Every identity in (7.1) holds in $\mathbf{W}$. Let $\mathbf{Z}$ be the variety they define. These identities give $\mathbf{W}\subseteq\mathbf{Z}\subseteq\mathbf{A}$. To prove the reverse inclusion, we derive every identity of $\mathbf{W}$ in $\mathbf{Z}$. Every doubled image of a member of $Z$ belongs to $V$. Since $d$ is a homomorphism, evaluating $P$ at doubled elements gives $2P$ at the original elements. The identities $2\Sigma_V$ imply $P \approx Q$ on each doubled image whenever $2P \approx2Q$ occurs in the basis. The protected identities in the basis give

By Lemma 5.2, $\mathcal{G}_Z$ is a fully invariant band theory. It contains $\Sigma_G$ because their protected versions occur in (27), and contains the band theory they generate.

Assume $W \models P \approx Q$ and expand both terms as polynomials. Put $X = \mathrm{c}(P) \cup\mathrm{c}(Q)$. By Lemma 4.3, their count functions and isolated contents agree. If $\nu_P(X) = 1$, both sides are single words, and their identity follows from $\Sigma_U$. It remains to consider polynomials with at least two occurrences on each side.

The common doubled part. Let

By Proposition 4.5, the following identities hold in $Z$:

where $\mathcal{I}$ is the common set of isolated contents. The polynomial $C$ is nonempty. Every isolated content is proper in $X$, since otherwise all occurrences of its side would be supported in it.

Keep $C$ fixed. Starting from the first expression in (29), apply only transformations described below that we have already proved in $Z$, and call the resulting polynomial $P_0$. Let $Q_0$ remain the second expression. The identity $P_0 \approx Q_0$ holds in $W$, since $W \subseteq Z$ and the original identity holds in $W$. Put $R = P_0 + Q_0$. In $V$ we have $R \approx P \approx Q$. If a word $h$ has content $X$ and $V \models R + h \approx R$, then

For the first statement, use the chosen free-band representative: $\nu_P(X) = 2$, and $V \models R \approx P$ gives the absorption condition in (12). For the second, let $\overline{h}$ be the representative of $h$ in $B_X$. The sum $C$ contains $2\overline{h}$, and $h \approx\overline{h}$ holds in every band. Adding $2h$ changes $2\overline{h}$ to $4\overline{h}$, which equals $2\overline{h}$. This coefficient reduction proves $C + 2h \approx C$ in $Z$ using its semiring and band identities. The absorption test in $V$ identifies a summand already in $C$; this argument does not assume $P_0 \approx Q_0$ in $Z$.

To choose $g$, multiply all word occurrences of $R$. The resulting word has content $X$, and $R$ absorbs it in $V$ because it absorbs every factor and their product. The content remains $X$ at every stage: $C$ contains the word obtained initially by multiplying all occurrences of $P$ and $Q$, using $V \models P \approx Q.

Alignment of the first-occurrence words. Fix a common isolated content $D$ and write $u,v$ for its current occurrences in $P_0,Q_0$. Suppose their first-occurrence words differ. If their first letters differ, Lemma 6.1 shows that $H_{\ell}\in\Gamma_W$. Apply it as in Lemma 6.2(i), with $a$ the first letter of $v$. Its undoubled words are $u,au$. The polynomial $R$ absorbs each instantiated guard word, and each has content $X$. By (30), the instantiated guard polynomial, say $C'$, satisfies $C+C'\approx C$. The guarded identity gives, in $Z$,

We add the other isolated summands to both sides.

If the first letters agree, let $p_0$ be the longest common proper prefix of $\ell(u),\ell(v)$, and let $a$ be the next target letter from $\ell(v)$. Lemma 6.1 gives $J_{\ell}\in\Gamma_W$. Choose an actual prefix $p$ as in Lemma 6.2(ii). That lemma and (30) allow the same calculation to replace $C+u$ by $C+pau$ in $Z$.

Each replacement lengthens the common first-occurrence prefix with $v$. It preserves content $D$ and the last-occurrence word, since the former word $u$ remains as a suffix. No occurrence in the fixed doubled part is supported in an isolated $D$: a word $q\in D$ with $c(q)\subseteq D$ would give $\nu_P(D)=2$. The replacement preserves isolation. Since the current identity $P_0\approx Q_0$ still holds in $W$, every guarded identity needed at the next step belongs to the selected set $\Gamma_W$. After finitely many steps the first-occurrence words agree.

Alignment of the last-occurrence words. Apply the reverse argument, using $H_r,J_r$. These replacements leave the whole current word as a prefix and preserve the first-occurrence word just obtained. After finitely many steps the resulting word $u^*$ satisfies

These transformations hold in $Z$ and retain the same $C$.

Replacement of the remaining word. The current identity $P_0\approx Q_0$ holds in $W$, and its occurrences $u^*,v$ remain isolated with content $D$. Theorem 5.4 and (31) imply $u^*\equiv_{\Gamma_W}v$. By (28), the protected identity

holds in $Z$. Choose an absorbed full-content word $g$ for the current $R$. Lemma 6.3 and (30) show that $C$ absorbs both $2u^*gu^*$ and $2vgv$. Substituting $\omega=g$ and adding $C$ yields

This replaces the isolated occurrence without changing $C$ or the other isolated contents.

Repeating this replacement for each of the finitely many $D\in\mathcal{I}$ makes the expressions in (29) identical; if $\mathcal{I}$ is empty, both already equal $C$. The completed derivation proves $Z\models P\approx Q$, giving $\operatorname{Id}(W)\subseteq\operatorname{Id}(Z)$ and $Z=W$.

Proof. A subvariety containing $T$ has the finite basis in Theorem 7.1. A subvariety not containing $T$ is contained in $J$ by Proposition 4.2, and is finitely based by Proposition 3.6. □

Proof of Theorem 1.1. The varieties $R$ and $A$ are hereditarily finitely based by Proposition 3.4 and Corollary 7.2. One identity defines each relative to $\mathrm{MI}$. Proposition 3.1 gives the required decomposition for every subvariety of $\mathrm{MI}$. Lemma 3.2 proves that $\mathrm{MI}$ is hereditarily finitely based. Corollary 1.2 follows by applying this conclusion to $\operatorname{Var}(S)$. □

An algebra generating a locally finite variety is called inherently nonfinitely based if it belongs to no finitely based locally finite variety of the same signature. The next corollary follows from the local finiteness of $\mathrm{MI}$ in Lemma 2.2, independently of Theorem 1.1.

Proof. The defining axioms give a finite basis for $\mathrm{MI}$, which is locally finite by Lemma 2.2. Every multiplicatively idempotent semiring belongs to a finitely based locally finite variety of the same signature. □

Remark 7.4. Theorem 1.1 proves that every subvariety of $\mathrm{MI}$ has a finite identity basis. The construction in (7.1) uses bases for the multiplicative band theory, the doubled image and the protected band theory, together with a subset of four fixed guarded identities. The auxiliary bases depend on the chosen subvariety. The proof gives no uniform bound on the number of variables, does not supply individual bases for every subvariety, and does not describe the full subvariety lattice of $\mathrm{MI}$.

Declaration of AI use

The authors used OpenAI ChatGPT and Codex for exploratory mathematical checks, manuscript organization, and language editing. They reviewed and edited the resulting content and take full responsibility for this article. School of Mathematical Sciences, Chongqing University of Technology, Chongqing 400054, P.R. China

Email address: [email protected]

School of Mathematical Sciences, Chongqing University of Technology, Chongqing 400054, P.R. China

Email address: [email protected]